Suppose a map is drawn using only lines that extend to infinity in
both directions; are two colors sufficient to color the countries so
that no pair of countries with a common border have the same color?

Consider seating n dinner guests at k tables of l (lower case L) settings
each, therefore n = k l , for m courses so that no guest shares a table
more than once with any other guest. Equivalently, consider n players to
be divided into k teams of l players for m rounds of a contest. No player
may, more than once, be on the same team as any other player. 1. What is
the maximum value of m, as a function of k and l? 2. How could one
systematically specify the seating arrangement for the m courses?

Prove: Assume that all points in the real plane are colored white or
black at random. No matter how the plane is colored (even all white or
all black) there is always at least one triangle whose vertices and
center of gravity (all 4 points) are of the SAME color.